Lemma 4.4 ([1]}, Proposition 6.9) 
For any \(\ell >0\) , there is a constant \(D^{\prime }_{\ell }\)  with the following property.
Fixed \((\omega ,x)\in \Gamma \) .
Then for any \(z^1,z^2\in \operatorname{graph}g_{0}\)  with \(|z^1-z^2|^{\prime }_{0}\le \delta ^{\prime }_1 (D^{\prime }_{\ell (0)})^{-1}\)  and any \(n\in \mathbb {N}\)  we have that
\(\bigg | \log \dfrac{\det ^{\prime }(D\widetilde{f}^n_{-n}(z^1_{-n})|E^1_{-n})}{\det ^{\prime }(D\widetilde{f}^n_{-n}(z^2_{-n})|E^2_{-n})} \bigg |\le D^{\prime }_{\ell (0)}|z^1-z^2|^{\prime }_{0},\) 
